Methods for evaluating formations using NMR and other logs

ABSTRACT

Methods that allow in-situ calculations of critical petrophysical parameters including, but not limited to, φ t , S xot , Q v , F, and R w  are provided. Also, NMR clay bound water may be used to estimate a continuous Q v . In combination with other resistivity logs, such as SP, R xo  and R deep , R w  can be determined. With the exception of the saturation exponent n, all Archie parameters and other computational equivalents are continuously determined directly from well logs. The methods therefore allow S w  to be determined more accurately, which leads to improved estimation of hydrocarbon reserves. The method is extended to complex lithology with additional tools. In complex lithology, permeability estimation is also improved using a method that estimates bound fluid volume. Also, an uncertainty analysis of the gas-corrected parameters is also provided.

CROSS REFERENCE TO RELATED APPLICATIONS

This is a continuation-in-part of copending U.S. patent application Ser.No. 08/873,981, filed Jun. 12, 1997, which claims the benefit of U.S.provisional application No. 60/042,059, filed Apr. 9, 1997.

FIELD OF THE INVENTION

This invention relates to methods for evaluating a subsurface formation.More particularly, this invention relates to methods for determiningparameters that characterize a formation using nuclear magneticresonance ("NMR") logging data, especially in combination with othertypes of logging data.

BACKGROUND OF THE INVENTION

The economic value of a formation containing hydrocarbons depends on theamount of oil or gas contained in a unit volume of a subsurfacereservoir, which, among other things, is a function of its porosity andits hydrocarbon saturation. Total porosity φ_(t) of a formation is thefraction of the formation per unit volume occupied by pore spaces.Hydrocarbon saturation S_(h) is a fraction of the pore volume filledwith hydrocarbons. In addition to porosity φ_(t) and hydrocarbonsaturation S_(h), permeability k of a formation indicates the ease withwhich a fluid (e.g., hydrocarbons) flows through, and can be removedfrom, the formation. Although a large porosity usually corresponds to alarge permeability, pore size, shape, and continuity also influencepermeability.

There are many well-known models that allow the calculation ofsaturation from well logs. In shaly formations, the preferred models arethe Waxman-Smits model (See, e.g., M. H. Waxman and L. J. M. Smits,"Electrical Conductivities in Oil-Bearing Shaly Sands," Society ofPetroleum Engineers 42nd Annual Fall Meeting, (Oct. 1-4, 1967), and theDual Water model (See e.g., C. Clavier, G. Coates, and J. Dumanoir, "TheTheoretical and Experimental Bases for the `Dual Water` Model for theInterpretation of Shaly Sands," Society of Petroleum EngineersTransactions 6859 (1977) (hereinafter, "Clavier et al."). Both modelsrely on the cation exchange capacity per unit volume Q_(v) and theformation factor F, which are not often measured downhole nor inferreddirectly from logging measurements.

NMR is based on the fact that the nuclei of many elements have angularmomentum (hereinafter, "spin") and a magnetic moment. Nuclear spinsalign themselves along an externally applied static magnetic field andobtain an equilibrium condition. This equilibrium can be disturbed by apulse of an oscillating magnetic field, which tips the spins away fromthe static field direction. The degree to which the spins are tipped isunder the control of the experimenter as explained below.

After tipping, two things occur simultaneously. First, the spins precessaround the static field at the Larmor frequency (ω_(o) =γB₀), where B₀is the strength of the static field and γ is the gyromagnetic ratio, anuclear constant. Second, the spins return to the equilibrium conditionaccording to a decay time known as the "spin-lattice relaxation time" orT₁. T₁ is controlled by the molecular environment and is typically tento one thousand milliseconds for water in rocks.

Also associated with the spin of molecular nuclei is a second relaxationtime known as "spin-spin relaxation time" or T₂. At the end of a ninetydegree tipping pulse, all the spins point in a common directionperpendicular to the static field, and they precess at the Larmorfrequency. However, small inhomogeneities in the static field due toimperfect instrumentation or microscopic material heterogeneities causeeach of the nuclear spins to precess at a slightly different rate.Therefore, after some time, the spins will not precess in unison, thatis they will dephase. When dephasing is due to static fieldinhomogeneity of the apparatus, the dephasing time is called T₂ *. Whenthe dephasing is due to properties of the material, the dephasing timeis called T₂.

T₂ can be several seconds in an unconfined low viscosity liquid such aswater, and as short as ten microseconds in a solid. Liquids confined inthe pores of rocks present an intermediate case where T₂ is in the rangeof tens to hundreds of milliseconds, depending on various factors, suchas pore size and fluid viscosity.

A known method for measuring T₂ is called the Carr-Purcell-Meiboom-Gill("CPMG") sequencing method. In solids, where T₂ is very short, T₂ can bedetermined from the decay of a detected signal after a ninety degreepulse. However, for liquids where T₂ *<<T₂, the free induction decaybecomes a measurement of the apparatus-induced inhomogeneities. Tomeasure the true T₂ in such liquids, it is necessary to cancel theeffect of the apparatus-induced inhomogeneities.

This cancellation is achieved by applying a sequence of pulses. Thefirst pulse is a ninety degree pulse that causes the spins to startprecessing. After the spins have begun precessing, a one hundred eightydegree pulse is applied to keep the spins in the measurement plane, butto cause the spins which are dispersing in the transverse plane toprecess in the reverse direction, thereby refocusing the spins. Byrepeatedly reversing and refocusing the spins by one hundred eightydegree pulses, a series of "spin echoes" occur. This succession of onehundred eighty degree pulses, after the initial ninety degree pulse, isthe Carr-Purcell sequence which measures the irreversible dephasing(i.e., T₂) due to material properties. Meiboom and Gill devised amodification to the Carr-Purcell pulse sequence such that, after thespins are tipped by ninety degrees and start to dephase, the carrier ofthe one hundred eighty degree pulses relative to the carrier of theninety degree pulse. As a result, any error that occurs during an evenpulse of the CPMG sequence is canceled out by an opposing error in theodd pulse. A detailed explanation of NMR principles and pulse sequencesis described in Freedman U.S. Pat. No. 5,291,137.

Unfortunately, the presence of gas in rock pores adversely effects thederivation of total formation porosity φ_(t). See, e.g., RobertFreedman, Austin Boyd, Greg Gubelin, Donald McKeon, and Chris Morriss,"Measurement of Total NMR Porosity Adds New Value to NMR Logging," PaperO, Transactions of the Society of Professional Well Log Analysts 38^(th)Annual Logging Symposium (1997).

For example, NMR-derived total porosities φ_(nmr) are generallyunderestimated when gas is present in the zone being measured. At leasttwo effects may be responsible for the underestimation of φ_(t). Thefirst effect is related to an abnormally low hydrogen index of gas. Thelow index effect is familiar to log analysts because it also causesneutron tool porosities to be reduced in gas zones. The second effect isrelated to insufficient polarization of the gas. The insufficientpolarization effect occurs because reservoir gas has longitudinalrelaxation times T₁ that are in the range from between 3 and 6 secondsat normal reservoir conditions. Because T₁ is so long, the time requiredto fully polarize reservoir gas is on the order of ten seconds usingconventional pulse sequences, such as the Carr-Purcell-Meiboom-Gill("CPMG") sequences. Unfortunately, a ten second wait time is generallyimpractical for routine logging operations because it results in veryslow logging speeds.

Many previously published methods for using NMR data to detect andquantify hydrocarbons in formations are "NMR-only" methods. That is,these methods use NMR data alone to derive hydrocarbon-related andporosity-related parameters. Most of these methods are based on conceptsintroduced by Akkurt et al., who recognized that the differences betweenthe NMR properties of water and non-wetting hydrocarbons in porous rocksprovides a means for distinguishing formation fluids into gas, oil, andwater volumes. R. Akkurt, H. J. Vinegar, P. N. Tutunjian, and A. J.Guillory, "NMR logging of natural gas reservoirs," Paper N, Transactionsof the Society of Professional Well Log Analysts 36th Annual LoggingSymposium (1995).

In the same paper, Akkurt et al. introduced a detailed method foridentifying and typing hydrocarbons. That method is called theDifferential Spectrum Method (hereinafter, "DSM"). Later, an improvementto the DSM method, known as Time Domain Analysis (hereinafter, "TDA"),was developed by M. G. Prammer, E. D. Drack, J. C. Bouton, J. S.Gardner, G. R. Coates, R. N. Chandler, and M. N. Miller, "Measurementsof clay-bound water and total porosity by magnetic resonance logging,"SPE Paper 35622, presented at the Society of Petroleum Engineers AnnualTechnical Conference and Exhibition (1996).

The DSM and TDA methods were both developed for use with tools having afixed magnetic field gradient (such as the tool available under thetrademark MRIL®, by Numar Corporation, of Malvern, Pa.). More recently,another NMR-only method of detecting gas, known as the Echo Ratio Method(hereinafter, "ERM"), was developed by Flaum et al. C. Flaum, R. L.Kleinberg, M. D. Hurlimann, "Identification of gas with the CombinableMagnetic Resonance tool (CMR*)," Paper L, Transactions of the Society ofProfessional Well Log Analysts 37th Annual Logging Symposium (1996). ERMuses a CMR tool which has a saddle point distribution of magnetic fieldgradients. ERM identifies gas using apparent diffusion constantscomputed from the ratios of two T₂ -decay curves acquired with differentecho spacings.

These NMR-only methods for calculating porosity and other parametershave various disadvantages. First, these methods work best with a toolthat has a fixed or saddle point distribution of magnetic fieldgradients. Thus, these methods are limited by the type of NMR tool usedto acquire data. Second, the NMR-only methods (e.g., ERM) may requiredata from two NMR measurements having different CPMG sequences. Third,the NMR-only methods require that the gas be appreciably polarized,which means long wait times and slow logging speeds. And fourth, totalporosity derivations from NMR-only techniques tend to be computationallycomplex.

The presence of gas also adversely effects the calculation ofdensity-derived total porosity φ_(density). Unlike NMR-derived totalporosity φ_(nmr), which underestimates true total porosity,density-derived total porosity overestimates true total porosity whengas is present in the formation. Thus, in a gas bearing zone, φ_(nmr)will be less than φ_(density) and the difference between the twoporosity logs will be proportional to the gas saturation in the zones.The difference effect is analogous to the "neutron-density" crossovereffect in gas zones. The same effect can occur when there is gascondensate or light oil in the formation. However, the magnitude of theeffect is reduced. The use of neutron-density logs for gas detection isnot reliable because the effects of shale and thermal neutron absorberson the neutron-density log response can totally suppress the crossovereffect. Also, neutron-density-derived total porosities can be inaccuratebecause of mineralogy effects on the neutron tool response.

Furthermore, conventional calculation of water saturation in shalyformations require knowledge of the formation factor F and cationexchange capacity per unit volume Q_(v). Obtaining this knowledgerequires core sample measurements. Such core sample measurements,however, are inconvenient, time-consuming, and costly because theyrequire that core samples be brought to the surface and analyzed,usually at an off-site laboratory. And, the cost generally scales withthe number of core samples analyzed, which at times can be very large.Therefore, immediate on-site valuation of Q_(v) and F are precludedusing conventional evaluation techniques.

In view of the foregoing, it is an object of this invention to providemethods for accurately determining gas-corrected flushed zone and virginzone parameters that characterize zones in a formation, even a shaly orgas bearing formation.

It is also an object of this invention to provide methods that allowimmediate on-site valuation of formations, without performing upholecore sample analysis.

It is also an object of this invention to provide methods thataccurately determine such parameters using nearly any conventional NMRtool, including fixed gradient tools and saddle point tools that have adistribution of magnetic field gradients.

It is yet another object of this invention to provide methods fordetermining gas-corrected total porosity and flushed zone gas saturationby combining NMR and density log measurements.

It is yet a further object of this invention to combine NMR measurementswith other open hole logs to determine critical petrophysicalparameters, such as virgin formation hydrocarbon saturation andpermeability, needed in the estimation of hydrocarbon reserves andproducibility.

It is yet an additional object of this invention to estimate theuncertainty of the magnitudes of the petrophysical parameters determinedin accordance with this invention.

SUMMARY OF THE INVENTION

These and other objects of the invention are accomplished in accordancewith the principles of the invention by providing methods that allowin-situ estimations of critical petrophysical parameters including, butnot limited to Q_(v), F, and R_(w), even in the difficult case of shalyand gas bearing formations. The method can also be used to providepermeability and producibility answers.

Furthermore, NMR clay bound water may be used to estimate a continuousQ_(v). With other resistivity logs such as SP, R_(xo), R_(deep),continuous F and R_(w) can be determined. With the exception of thesaturation exponent n, all Archie parameters are continuously determineddirectly from well logs, including any computational equivalents ofthese parameters. As used herein, a computational equivalent is anyparameter immediately derivable from a parameter that has beendetermined according to this invention. The method thus allows S_(w) tobe determined more accurately, which leads to improved estimation ofhydrocarbon reserves. The methodology is extended to complex lithologywith additional tools.

In accordance with this invention, a method for characterizing agas-bearing formation traversed by a borehole is provided. The methodincludes (1) computing an NMR-derived total porosity φ_(nmr) and adensity-derived total porosity φ_(density), (2) determining agas-corrected total porosity φ_(t) using the φ_(nmr) and theφ_(density), (3) determining a gas-corrected total water saturationS_(xot) in the flushed zone using the φ_(nmr) and the φ_(density), and(4) determining resistivity parameters, X_(mf) and m, using Archie'sequation extended for shaly formations: ##EQU1## n is a saturationexponent, m is a model-dependent cementation exponent, C_(xo) is aconductivity in the flushed zone, C_(xo) being equal to 1/R_(xo), whereR_(xo) is a flushed zone resistivity, C_(mf) is a mud filtrateconductivity, C_(mf) being equal to 1/R_(mf), where R_(mf) is aresistivity of the mud filtrate, and where X_(mf) is a model-dependentclay conductivity. The parameters that appear in Equation (1) aresometimes referred to as Archie parameters. φ_(t), X_(mf), and m arethen used in conjunction with a true resistivity R_(true) to compute awater saturation S_(wt) in the virgin zone according to: ##EQU2## withC_(true) =1/R_(true), where R_(true) is the virgin zone resistivity, andC_(w) =1/R_(w), where R_(w) is the water resistivity.

In accordance with another aspect of this invention, a method forcharacterizing a formation traversed by a borehole using NMR datawithout density data is also provided. The method includes in a firststep receiving NMR data characterizing the flushed zone. The NMR datapreferably includes a T₂ distribution P(T₂). Then, in a second step,clay bound water volume Vbound is determined substantially according to:##EQU3## where T₂ min is a minimum T₂ for clay bound water and T₂ max isa maximum T₂ for clay bound water. And, in a third step, a cationexchange capacity per unit total pore volume Q_(v) is determined using aclay bound water saturation S_(wb) model.

In yet another aspect of this invention, a method for characterizing aformation traversed by a borehole using the spontaneous potential isprovided. The method includes receiving SP logging data and using thatdata to determine R_(w) or a cation exchange capacity per unit totalpore volume Q_(v) with an electrochemical potential model, such as theone detailed in L. J. M. Smits, "SP Log Interpretation in Shaly Sands,"Society of Petroleum Engineers Journal, vol. 8, pp. 123-136 (1968).Preferably, the method involves at least a two step process. First, theLaplace Equation ∇σ_(c) ·∇V=0 is solved to determine the SP source,where σ_(c) is a conductivity and V is a potential everywhere in theformation. Next, the SP source integral Equation (33) is solved forR_(w) or Q_(v).

And, according to yet a further aspect of the invention, a method fordetermining the bound fluid volume BFV of a formation with complexlithology traversed by a borehole is provided. The method includesreceiving NMR data characterizing a flushed zone of the formation anddetermining the bound fluid volume BFV of the formation by summing theBFV_(i) constituents weighted by their respective constituent volumesV_(i), where i is an index denoting different constituents.

Furthermore, according to an additional aspect of this invention, amethod for analyzing the uncertainty of certain resistivity parametersis provided. The method includes calculating the variance of agas-corrected petrophysical parameter substantially according to:##EQU4## where ƒ is a parameter that is a function of n variables x_(n),σ² (ƒ) is a variance of ƒ, and x_(i) * is a best estimate for each ofthe n variables.

Further features of the invention, its nature and various advantageswill be more apparent from the accompanying drawings and the followingdetailed description of the preferred embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a simplified schematic representation of a horizontalcross-section of a permeable formation.

FIG. 2 is a flow chart of steps for carrying out a first illustrativeembodiment of the method for characterizing a gas-bearing formationtraversed by a borehole.

FIG. 3 is a flow chart of steps for carrying out a second illustrativeembodiment of the method for characterizing a gas-bearing formationtraversed by a borehole.

FIG. 4 is a flow chart of steps for carrying out a third illustrativeembodiment of the method for characterizing a gas-bearing formationtraversed by a borehole.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 is a simplified schematic representation of a horizontalcross-section of a permeable formation 100 in which a borehole has beendrilled. During drilling, the hydrostatic pressure of mud column 120 inborehole 110 is usually greater than the pore pressure of formation 100.The resistivity of the mud in the borehole is called mud resistivityR_(m). The pressure differential between mud column 120 and formation100 forces the mud-filtrate that makes mud column 120 into formation100, and the solid particles of the mud are deposited on the boreholewall after they form mudcake 130, which has resistivity R_(mc). Innerdiameter 132 of mudcake 130 is normally measured with calipers and thusis sometimes referred to as "Cali." Outer diameter 134 is determined bythe outer diameter of the bit used to drill the borehole.

In the radial zone close to borehole 110, most of the original formationwater and some hydrocarbons may be flushed away by the filtrate. Thiszone is referred to as flushed (or invaded) zone 140, which hasresistivity R_(xo). The resistivity of the mud filtrate in flushed zone140 is called mud filtrate resistivity R_(mf). Outer diameter 150 offlushed zone 140 is sometimes referred to as "Di."

Beyond flushed zone 140, the displacement of the formation fluids by themud filtrate is less complete. The radial extent of flushed zone 140depends on many factors including the type of drilling mud, theformation porosity, the formation permeability, the pressuredifferential, and the time since the formation was first drilled. Theundisturbed formation beyond the flushed zone is referred to asuncontaminated or virgin zone 160, which has true resistivity R_(true).The water in virgin zone 160 has resistivity R_(w).

The electrical resistivity of a substance is its ability to impede theflow of electrical current through the substance. Electricalconductivity is the reciprocal of resistivity. Resistivity measurementshave been employed, singly and in combination, to determine R_(true) invirgin zone 160. They are also used to determine R_(xo).

The resistivity of clean, water-bearing formations (i.e., one containingno appreciable amount of clay and no hydrocarbons) is proportional tothe resistivity R_(w) of brine when saturated. The constant ofproportionality is called the formation resistivity factor, F. Archieproposed a formula relating porosity φ to the formation factor F(F=α÷φ^(m)), where α is a proportionality factor and m is a cementationexponent (see G. E. Archie, "The Electrical Resistivity as an Aid inDetermining Some Reservoir Characteristics," J. Pet. Tech., Vol. 5, No.1, (January 1942) and G. E. Archie, "Classification of CarbonateReservoir Rocks and Petrophysical considerations," Bull., AAPG, Vol 36,No. 2, (February 1952)).

Assuming that the rocks in the formation are perfect insulators whendry, the resistivity of a formation containing oil or gas and water is,among other things, a function of F, R_(w), and S_(w), where S_(w) isthe fraction of the pore volume occupied by formation water. FromArchie's equation, water saturation S_(w) of a clean formation can beexpressed in terms of its true resistivity R_(true) as: ##EQU5## where nis a saturation exponent, which is usually approximated as 2. Water (mudfiltrate) saturation S_(xo) of flushed zone 140 can also be expressed ina similar fashion as: ##EQU6## S_(xo) of flushed zone 140 is equal to1-S_(hr), where S_(hr) is the residual hydrocarbon saturation in flushedzone 140.

Some rocks in the formation, such as clays and shales, are not perfectinsulators when dry. For example, clays and shales have substantialconductivities. Formations that are substantially shaly are usuallyreferred to as shaly formations, and are often considered difficultformations to evaluate using conventional logging techniques. Becauseall logging measurements are influenced by the presence of shale,corrections for shale content are required for an accurate picture ofthe formation.

In accordance with this invention, at least three methods forcharacterizing gas-bearing formations using NMR data, density data, andelectrical data are provided. In a first method, Q_(v) is computed froman NMR tool measurement (see FIG. 2). In a second method, Q_(v) iscomputed from a shallow electrical tool measurement (see FIG. 3). And,in a third method, Q_(v) is computed from a spontaneous potentialmeasurement SP (see FIG. 4).

FIG. 2 shows first method 200 for characterizing a formation accordingto this invention. Method 200 at least includes, in step 210, computingat least φ_(nmr), φ_(density), and φ_(bound) of a flushed zone; in step215, computing at least φ_(t) and S_(xot) using φ_(nmr) and φ_(density)in the flushed zone; in step 220, computing a formation factor F; instep 209, making electrical tool measurements; in step 217, computingR_(w) ; in step 230, computing S_(wt) ; and in step 240, estimatingreserves and producibility of the formation. The method involvesdetermining X_(mf) and m using: ##EQU7## where n is a saturationexponent, m is a model-dependent cementation exponent, C_(xo) is aconductivity in the flushed zone, C_(xo) being equal to 1/R_(xo), whereR_(xo) is a flushed zone resistivity, C_(mf) is a mud filtrateconductivity, C_(mf) being equal to 1/R_(mf), where R_(mf) is aresistivity of the mud filtrate, and where X_(mf) is a model-dependentclay conductivity. Equation (7) is sometimes referred to as Archie'sequation as extended to shaly formations. It is understood that each ofthe variables above may correspond to one point in the flushed zone ofthe formation (e.g., a single depth) or to a set of points (e.g., adepth profile) in the flushed zone of the formation.

Before φ_(nmr) is computed in step 210, NMR data that characterizes theformation can be measured, as shown in step 205. In step 205, a firstportion of the earth formation is measured with a nuclear magneticresonance tool. This preferably involves measuring a first portion ofthe earth formation with a down-hole nuclear magnetic resonance tool toobtain NMR data. Any type of down-hole NMR-logging tool can be used(such as, for example, the tools sold under the trademark CMR®, bySchlumberger Technology Corporation, of Houston, Tex. or under thetrademark MRIL®, by Numar Corporation, of Malvern, Pa.).

Once φ_(nmr) is computed, it is preferably stored in a memory unit. Thememory unit may be downhole or uphole. Alternatively, φ_(nmr) may betransmitted directly to a processor for use in determining φ_(t) andS_(xot) in step 215.

φ_(density) is preferably computed from density data that characterizesthe formation. Therefore, in step 205, a second portion of the earthformation is also measured with a density tool, preferably with adown-hole density tool. Any type of down-hole density tool can be used,including a high-energy gamma-gamma radiation tool.

Preferably, the first and second portions measured in step 205 aresubstantially the same. This means that, at a particular longitudinal(e.g., vertical) position along the borehole, the radial (e.g., lateral)depths of evaluation of the NMR and density tools are well matched. Anexample of a well matched pair of tools is a CMR tool and a high energygamma-gamma radiation tool. See D. V. Ellis, Well Logging for EarthScientists (1987). Thus, in accordance with the principles of thisinvention, it is assumed that shallow-reading tools, such as NMR,Density, R_(xo), and EPT logging tools, evaluate substantially the samelongitudinal and radial location of flushed zone 140. Moreover, in orderto obtain virgin zone parameters in step 230, it is further assumed thatφ_(t) and F are substantially laterally invariant.

φ_(density) may be computed in step 210 from the density data measuredin step 205 using any conventional calculation method. Once computed,φ_(density) may be stored in a memory unit. This memory unit may be thesame memory unit used to store φ_(nmr), or a different memory unit, andmay be located downhole or uphole. Like φ_(nmr), φ_(density) may betransmitted directly to a processor for use in determining φ_(t) andS_(xot) in step 215 without being stored.

In step 215, the gas-corrected total porosity φ_(t) may be determinedsubstantially according to: ##EQU8## where (HI)_(g) is the HydrogenIndex of the gas, (HI)_(f) is the Hydrogen Index of the liquid phaseconsisting of a mixture of mud filtrate and formation water, P_(g) isthe gas polarization function, which is defined as 1-exp(-WT/T_(l),gas), where WT is the wait time for a pulse sequence andT_(l),gas is a gas longitudinal relaxation time (See R. Freedman, "GasCorrected Porosity from Density Porosity and CMR Measurements in `How toUse Borehole NMR,`" Oilfield Review, vol. 9, No. 2, pp. 54)(hereinafter, "Freedman"). λ is proportional to the density differencebetween the gas and liquid phases and is responsible for the gas effecton φ_(density). λ may be determined substantially according to: ##EQU9##where ρ_(f) is the density of the liquid phase, ρ_(ma) is the formationmatrix density, and ρ_(g) is the density of the gas. Computation ofφ_(density) in step 210 requires at least two inputs, including ρ_(ma)and ρ_(f). φ_(density) may be determined according to: ##EQU10## whereρ_(b) is the formation bulk density ρ_(b).

It should be understood by a person of ordinary skill in the art thatthe computation of φ_(t) and S_(xot) can be performed individuallythrough a series of intermediate steps, or by calculating anycomputationally equivalent parameter. Computation of φ_(t) and S_(xot)in step 215 requires several inputs, including, for example, ρ_(g),T_(l),gas, (HI)_(g), (HI)_(h), ρ_(f), and ρ_(ma), as shown in step 211.

When the formation is lithologically simple (i.e., the formation onlyincludes one principal type of rock), determination of the matrixdensity ρ_(ma) and other density-derived parameters, is straightforwardbecause the value of ρ_(ma) is usually well-known. However, when aformation includes two or more principal constituents (such as sandstoneand limestone), ρ_(ma) may be determined, in step 207, substantiallyaccording to: ##EQU11## where V_(i) is the volume and ρ_(i) is thedensity of a formation constituent i. V_(i) may be determined using oneor more logging techniques. ρ_(i) values for most constituents aregenerally known to a person of ordinary skill in the art. For example,the density of a formation of limestone ρ_(limestone) is known to beabout 2.71 g/cc and the density of a formation of sandstoneρ_(sandstone) is about 2.65 g/cc. Some of the techniques that may beused to obtain V_(i) include Thorium logging, Potassium logging, Neutronlogging, Sonic logging, Photoelectric logging, Elemental Yield logging,and any combination thereof.

In step 215, φ_(t) may be determined from Equation (8) substantiallyaccording to: ##EQU12## where the w is determined substantiallyaccording to: ##EQU13## where (HI)_(g), (HI)_(f), P_(g), and λ aredefined above. Therefore, when the formation has a complex lithology, amore accurate determination of w can be achieved by determining ρ_(ma)according to Equation (11).

Generally, in a gas reservoir, w has a value between about 0.55 andabout 0.65, and thus φ_(t) can be estimated using a value in that range.φ_(t) is preferably estimated by setting w equal to about 0.60. Thus,Equation (12) reduces to: ##EQU14##

In addition to determining φ_(t) in step 215, the gas-corrected totalwater saturation S_(xot) in the flushed zone, or a computationalequivalent, is computed. Preferably, S_(xot) is determined substantiallyaccording to: S_(xot) =V_(xot) /φ_(t), where V_(xot) is the total watervolume of the flushed zone. V_(xot) may be determined substantiallyaccording to: V_(xot) =φ_(t) -V_(g),xo, where V_(g),xo is the gas volumeof the flushed zone. V_(g),xo and S_(g),xo may be determinedsubstantially according to (See, Freedman): ##EQU15##

Next, one or more resistivity parameters are determined, including m, X,and a cation exchange capacity per unit total pore volume Q_(v).Preferably, m, X, and Q_(v) are all determined.

It is useful to provide a value for n and to determine a value forC_(xo). A commonly used value for n is 2. Determining C_(xo) may involvemeasuring C_(xo) with a shallow-resistivity tool. After n is providedand C_(xo) is determined, any saturation model can be used to determineformation factor F in step 220.

One model that can be used in accordance with this invention is theWaxman-Smits model M. H. Waxman and L. J. M. Smits, "ElectricalConductivities in Oil-Bearing Shaly Sands," Society of PetroleumEngineers 42nd Annual Fall Meeting held in Houston, Tex., Oct. 1-4),1967. According to the Waxman-Smits model, X=Q_(v) B, where: ##EQU16##where C_(w) is the conductivity of water and T_(C) is the temperature ofthe formation in degrees Celsius. In the flushed zone, water is the mudfiltrate and in the virgin zone, water is the formation water. Then, Xis computed accordingly. Furthermore, according to this model, ##EQU17##In combination, Equations (7) and (17)-(19) can be used by a person ofordinary skill in the art to determine any or all of the desiredresistivity parameters (e.g., Q_(v) and m_(ws)).

Another model that may be used in accordance with this invention is theDual Water model. (See, Clavier et al.) According to the Dual Watermodel: X=(C_(wb) -C_(mf)) S_(wb), where C_(mf) is the conductivity ofmud-filtrate, ##EQU18## where C_(wb) is the clay bound waterconductivity, S_(wb) is the clay bound water saturation determinedsubstantially according to: S_(wb) =α·V_(q) ·Q_(V), where α may bedetermined substantially according to: ##EQU19## and where γ is theactivity coefficient determined substantially according to: ##EQU20##Coefficients a₁, a₂, a₃, a₄, and a₅ are determined substantiallyaccording to:

    a.sub.i =b.sub.i T.sub.c.sup.3 +c.sub.i T.sub.c.sup.2 +d.sub.i T.sub.c +e.sub.i,                                                 (24)

where i=1, 2, 3, 4, and 5, respectively, m is a salinity of solvent,preferably in moles/kg, and the coefficients b_(i), c_(i), d_(i), ande_(i) are about:

    ______________________________________                                        i   b.sub.i    c.sub.i    d.sub.i  e.sub.i                                    ______________________________________                                        1   -6.1237e-11                                                                              +3.6490e-08                                                                              -1.2225e-06                                                                            +9.7432e-04,                               2   -3.1529e-08                                                                              +8.7540e-06                                                                              -1.3528e-03                                                                            -2.4460e-01,                               3   +1.5951e-08                                                                              -7.0447e-06                                                                              +1.0840e-03                                                                            +1.0514e-01,                               4   -1.0729e-09                                                                              +5.5435e-07                                                                              -1.0211e-04                                                                            +4.7400e-04, and                           5   +4.1937e-09                                                                              -2.1167e-06                                                                              +1.1317e-04                                                                            -3-6126e-02.                               ______________________________________                                    

m may be determined substantially according to: ##EQU21## Also, n is thesalinity, preferably in moles/l, which may be determined substantiallyaccording to: ##EQU22## where ppk is the salinity in parts per thousandand ρ_(f) is a fluid density of the liquid phase, which includes amixture of mud filtrate and formation water, preferably in g/cc. Vq maybe determined substantially according to:

    Vq=(4.97×10.sup.-6 T.sub.c.sup.2)-(1.94×10.sup.-3 T.sub.c)+0.342.                                           (27)

where T_(c) is the temperature in degrees Celsius. Also, ##EQU23## whereF_(dw) is the Dual Water formation factor,

    m.sub.dw =1.7762+0.3364(1-e.sup.-5.5035y)                  (29)

and y is previously defined by Equation (20). In combination, Equations(20)-(29) can be used by a person of ordinary skill in the art todetermine any or all of the resistivity parameters.

In step 230, a gas-corrected virgin zone parameter can be determined.The method preferably includes measuring a deep conductivity C_(deep)(or equivalently R_(deep), to calculate C_(true)) in the virgin zone,and determining a virgin zone water saturation S_(wt) substantiallyaccording to: ##EQU24## where C_(w) is the conductivity of water in thevirgin zone. C_(true) is preferably determined using a deep-resistivitymeasuring tool. C_(w) (or R_(w)) may be determined in step 217 usingdata obtained by electrical tool measurements in step 209 in combinationwith values computed for Q_(v) and S_(xot) in step 215. Inputs requiredfor that determination include R_(m), Cali, and Di, as provided in step212. After S_(wt) is determined in step 230, one can estimate reservesand producibility of the formation, such as by calculating hydrocarbonsaturation S_(hy) in the virgin zone substantially according to: S_(hy)=1-S_(wt).

Computation of φ_(bound) (also referred to as V_(bound)) in step 210 caninvolve, in a first step, receiving NMR data characterizing a flushedzone. The NMR data at least includes a T₂ distribution P(T₂). In asecond step, a clay bound water volume V_(bound) is determinedsubstantially according to: ##EQU25## T₂ min is a minimum T₂ for claybound water and T₂ max is a maximum T₂ for clay bound water. In a thirdstep, a cation exchange capacity per unit total pore volume Q_(v) isdetermined using a clay bound water saturation S_(wb) model. Q_(v) canbe calculated using a Hill-Shirley-Klein model substantially accordingto: ##EQU26## where φ_(t) is the total porosity of the formation, and nis the salinity (e.g., in meq/cc). φ_(t) is preferably determined fromthe combination NMR-density technique.

The integration limits of Equation (31) may be estimated with knownvalues. For example, T₂ min may be fixed at about 0.1 msecs and T₂ maxmay be fixed at about 3.0 msecs. However, as more knowledge about NMRclay bound water is gained, better estimates of T₂ min and T₂ max may beused to calculate V_(bound) in accordance with this invention. OnceV_(bound) is determined, S_(wb) may be determined substantiallyaccording to: S_(wb) =V_(bound) /φ_(t), where φ_(t) is a total porosityof the formation. Although φ_(t) may be an NMR-derived total porosityφ_(nmr), φ_(t) may also be derived from any other logging toolmeasurement and is best derived from the combination NMR-densitytechnique.

Another method 300 for characterizing a formation according to thisinvention, as shown in FIG. 3, includes the following steps. In step310, computing at least φ_(nmr) and φ_(density) of a flushed zone; instep 309, making electrical tool measurements; in step 315, computing atleast φ_(t), S_(xot), Q_(v), and F in the flushed zone; in step 317,computing R_(w) ; in step 330, computing S_(wt) ; and in step 340,estimating reserves and producibility of the formation.

Many of the steps of method 300, shown in FIG. 3, are the same as thesteps of method 200, shown in FIG. 2. For example, before φ_(nmr) andφ_(density) are computed in steps 210 or 310, NMR data and density datathat characterize the formation are preferably obtained by measurementwith an NMR tool and a Density tool, as shown in steps 205 and 305.Also, the inputs used in steps 210, 215, 217 and 230 are substantiallythe same as the inputs used in steps 310, 315, and 317. As describedabove, the primary difference between method 200 and method 300 is thatmethod 200 computes Q_(v) from NMR data and method 300 computes Q_(v)from shallow resistivity data R_(xo).

The third method for characterizing a formation according to thisinvention, as shown in FIG. 4, involves computing Q_(v) from spontaneouspotential measurement SP. Method 400 includes, in step 410, computing atleast φ_(nmr) and φ_(density) of a flushed zone; in step 415, computingat least φ_(t) and S_(xot) in the flushed zone; in step 409, makingelectrical tool measurements; in step 417, computing Q_(v) ; in step420, computing F; in step 430, computing S_(wt) ; and in step 340,estimating reserves and producibility of the formation.

The method involves solving ∇σ_(c) ·∇V=0, where σ_(c) is theconductivity and V is a potential everywhere in the formation. In orderto solve the equation, at least two boundary conditions are used. Thefirst boundary condition is V₂ -V₁ =SSP and the second boundarycondition is J₂ -J₁ =0. SSP is a strength of an electro-chemicalpotential at the interface between the flushed zone and the virgin zone,J is the electric current density at the interface, and the subscripts 1and 2 denote the flushed and virgin zones at the interface,respectively. SSP may be calculated with any conventional method such aswith a finite element method (See, e.g., M. Y. Chen, C. Cao Minh,"Determination of Continuously Varying R_(w) from SP," InternationalSymposium on Well Logging Techniques for Oilfield Development UnderWaterflooding, SPWLA, Beijing, China, September 1996), or adeconvolution method (See, e.g., J. R. Tabanou, G. Glowinsky, and G. F.Rouault, "SP Deconvolution and Quantitative Interpretation in ShalySands," SPWLA 28th Symposium, paper SS (1987)) or from publishedcorrection charts (See, e.g., F. F. Segesman,, "New SP CorrectionCharts," Geophysics, vol. 27, no. 6 (December 1962)).

SSP may be calculated by measuring the spontaneous potential SP formedbetween two points in the borehole, resistivity of the flushed zoneR_(xo), mud resistivity in the borehole R_(m), and the boreholecross-sectional area (i.e., ##EQU27## as well as resistivity in thevirgin zone, resistivity of the surrounding beds, and the position ofthe electrochemical potential.

Alternatively, SSP may be determined substantially according to:##EQU28## where k is the Boltzmann constant, T is the absolutetemperature of the formation, e is the electron charge, m is thesalinity (moles/kg), γ is the activity coefficient (see, e.g., Equations(23)-(26)), C₊ is the cation conductivity and C₋ is the anionconductivity, and C is the rock conductivity determined substantiallyaccording to: C=C₊ +C₋.

The cation conductivity C₊ may be determined substantially according to:##EQU29## Also, the anion conductivity C₋ may be determinedsubstantially according to:

    C.sub.- =S.sub.xot.sup.n φ.sub.t.sup.m (1-t)C.sub.f.   (35)

t is the cation transference number and C_(f) is the conductivity of thefluid at the interface. For NaCl solutions, t may be determinedsubstantially according to:

    t=0.374-0.125 log (m)-1.77e-3 log.sup.2 (m)+4.047e-4(T.sub.c.sup.0 -25)-8.22e-7(T.sub.c.sup.0 -25).sup.2,                    (36)

and C_(f) may be determined substantially according to: ##EQU30## γ maybe determined substantially according to Equations (23)-(25). Therefore,Equation (33) can be used to determine C_(w) knowing Q_(v) and S_(xot)from the aforementioned techniques, or to determine Q_(v) knowing C_(w)and S_(xot) from the aforementioned techniques.

Like method 300, many of the steps in method 400, shown in FIG. 4, aresubstantially the same as the steps in method 200, shown in FIG. 2. Forexample, the inputs provided in steps 207, 211, 212, and 213, and usedin steps 210, 215, 217 and 230, are substantially the same as the inputsused in steps 410, 415, and 317. As described above, the primarydifference between method 400 and the previous methods 200 and 300 isthat method 400 computes Q_(v) from spontaneous potential data SP. R_(w)is not computed (as in steps 217 and 317 of methods 200 and 300,respectively. Rather, R_(w) is provided in step 412 and used to computeQ_(v) in step 417.

A method for computing bound fluid volume BFV of a formation withcomplex lithology is now described. The method includes receiving NMRdata characterizing the flushed zone of the formation; and determiningthe bound fluid volume BFV of the formation. BFV is determined bysumming the BFV_(i) constituents weighted by their respectiveconstituent volumes V_(i), where i is the index denoting differentconstituents. The NMR data at least includes P(T₂), which is the T₂distribution.

More particularly, determining the BFV may be substantially accordingto: ##EQU31## where T₂ min is the minimum T₂, and T₂ cutoffi is thecutoff T₂ of constituent i. Although the most accurate determinationwould include every constituent present in the formation, a simplifieddetermination would only include the principal, or most abundant,constituents.

For example, if a formation is mainly formed from limestone andsandstone, the formation is said to have two principal constituents.Therefore, according to one aspect of this invention, the BFVdetermination using Equation (38) could be simplified to only includetwo terms--one for limestone and one for sandstone. That is, the BFV fora formation having two principal constituents can be expressed as:##EQU32## where V₁ is the volume of the first principal constituent andV₂ is the volume of the second principal constituent, T₂ cutoff1 is thecutoff T₂ of constituent 1, T₂ cutoff2 is the cutoff T₂ of constituent2, and as above, P(T₂) is the T₂ distribution.

The method may further include determining a permeability k of theformation according to a permeability model and the BFV calculatedaccording to this invention. That determination may use any knownpermeability model.

This method may further include determining a T₂ cutoff for the mixturesubstantially according to: ##EQU33## where T₂ min is the known minimumT₂, BFV is the known bound fluid volume for the formation determinedfrom Eq. 39, and P(T₂) is the T₂ distribution. T₂ min may be estimatedto be about 0.3 msecs. Of course, this estimate is used only as anexample and will depend on the particular formation being studied andthe accuracy of the calculation desired.

TABLE 1 summarizes the different determinable parameters using differenttool combinations:

                  TABLE 1                                                         ______________________________________                                        Tool                      NMR +    NMR + LDT +                                Combination                                                                           NMR    NMR + LDT  LDT + R.sub.xo                                                                         R.sub.xo  + R.sub.deep  + SP               ______________________________________                                        Determin-                                                                             .O slashed..sub.nmr                                                                  .O slashed..sub.nmr  BFV, k,                                                             .O slashed..sub.nmr  BFV, k,                                                           .O slashed..sub.nmr  BFV, k, .O                                               slashed..sub.t,                            able    and    .O slashed..sub.t, .O slashed..sub.e,                                                    .O slashed..sub.t, .O slashed..sub.e,                                                  .O slashed..sub.e, .O slashed..sub.g,xo                                       , Q.sub.v,                                 Parameters                                                                            BFV    .O slashed..sub.g,xo  and Q.sub.v                                                        .O slashed..sub.g,xo, Q.sub.v                                                          m, R.sub.w  and .O slashed..sub.g                                    and m                                               ______________________________________                                    

The table includes five types of tools that can be used in accordancewith the principals of this invention. However, it should be clear to aperson of ordinary skill in the art that other tools could besubstituted for, or used in addition to, these tools as desired. NMR isthe nuclear magnetic resonance tool; LDT is the logging density tool;R_(xo) is the shallow-resistivity tool for measuring flushed zoneresistivity; R_(deep) is the resistivity tool for measuring virgin zoneresistivity. The determinable parameters, in addition to the onesdescribed above and any computational equivalents, include φ_(e), whichis an effective porosity.

The parameters determined according to Equations (8), (15), and (16)depend on the NMR properties and bulk densities of the fluids in theformation, the formation matrix density, the measured formation bulkdensities, and the total NMR porosities. The NMR properties of thefluids and the fluid densities depend on fluid type, reservoirtemperature, and pressure. For bulk fluids these properties can beestimated from published charts and literature data. See, e.g., R.Akkurt, H. J. Vinegar, P. N. Tutunjian, A. J. Guillory, "NMR Logging ofNatural Gas Reservoirs," Paper N. Transactions of the Society ofProfessional Well Log Analysts 36th Annual Logging Symposium (1995); andR. L. Kleinberg and H. J. Vinegar, "NMR Properties of Reservoir Fluids,"The Log Analyst, (November-December, 1996). Furthermore, a recent papershows that the methane gas longitudinal relaxation times in rocks arereduced from their bulk values by surface relaxation. See, C. Straley,"An Experimental Investigation of Methane in Rock Materials," Paper AA,Transactions of the Society of Professional Well Log Analysts 38thAnnual Logging Symposium (1997). This effect adds additional uncertaintyto our estimation of in-situ NMR relaxation times of reservoir gas.

The inputs used by Equations (8), (15), and (16) are ρ_(b), ρ_(ma),ρ_(f), ρ_(g) , T_(l),gas, (HI)_(g), (HI)_(f), φ_(nmr), and WT. Theuncertainties in the outputs (e.g., see steps 215, 315, and 415) can becomputed from the uncertainties assigned to each of the inputs, whichdepend on the logging environment (e.g., see steps 211, 311, and 411).For example, in a shaly sand development well a log analyst or geologistmight reasonably assign a value to the formation matrix density thatassumes a small uncertainty (e.g., ρ_(ma) =2.65±0.03 g/cm³). In a shalysand exploration well with unknown mineralogy the formation matrixdensity might reasonably be assigned a greater uncertainty (e.g., ρ_(ma)=2.65±0.05 g/cm³). The input uncertainties usually reflect our lack ofdetailed knowledge of a particular parameter. There are alsouncertainties in measured log responses. These are due to measurementerrors and to statistical errors arising from random noise. For example,a formation bulk density tool measurement may have a total measurementuncertainty of ±0.01 g/cm³.

In light of these uncertainty factors, it would be desirable to estimatethe magnitude of the uncertainties of the resistivity parametersdetermined from Equations (8), (15), and (16).

Thus, in accordance with the principles of this invention, a method foranalyzing the uncertainty of resistivity parameters is provided. Themethod includes calculating the variance of a resistivity parametersubstantially according to: ##EQU34## where ƒ is a parameter that is afunction of n variables x_(n), σ² (ƒ) is a variance of ƒ, and x_(i) * isa best estimate for each of the n variables. The best estimates arepreferably the statistical expectation values of the variables. However,in practice these values are user-assigned inputs. Equation (41) assumesthat the uncertainties of all input parameters are statisticallyindependent and that third and higher order terms in the deviations (x₁-x_(i))* can be neglected. The variance σ² (x_(i)) in each input is thesquare of the uncertainty assigned to that input and the variance σ² (ƒ)is the square of the uncertainty in ƒ. See, R. Freedman and B. E.Ausburn, "The Waxman-Smits Equation for Shaly Sands: I. Simple Methodsof Solution: II. Error Analysis: The Log Analyst, at 11-23 (March-April,1985).

ƒ may be any calculable parameter, and ƒ is preferably a gas-correctedtotal porosity φ_(t), a gas volume of the flushed zone V_(g),xo, or aflushed zone gas saturation S_(g),xo. For calculation of these outputs,useful quantities N₁, N₂, a⁰, and D are defined substantially accordingto: ##EQU35## where (HI)_(g), (HI)_(f), P_(g), λ, φ_(density), andφ_(nmr) are defined as above.

When calculating the variance of φ_(t) according to Equation (41), thefollowing partial differential equation may be used: ##EQU36##

When calculating the variance of V_(g),xo according to Equation (41),the following partial differential equations may be used: ##EQU37##

The variance of S_(g),xo may also be directly calculated substantiallyaccording to Equation (41) or indirectly calculated substantiallyaccording to: ##EQU38## where σ² (φ_(t)) and σ² (V_(g),xo) arecalculated substantially according to Equations (41) in conjunction with(42)-(53) and (54)-(61), respectively.

The following examples use synthetic data to show that these calculatedoutputs are relatively insensitive to realistic input uncertainties.

HIGH POROSITY SHALY GAS SAND EXAMPLES

TABLE 2 contains synthetic data inputs used to compute the outputs shownin TABLE 3 for twelve high-porosity shaly gas sand examples. Theexamples illustrate the magnitude of the errors in φ_(t) and V_(g),xothat arise from realistic uncertainties in the inputs.

Examples 1-3 in TABLES 2 and 3 assume that the matrix density ρ_(ma) isknown to within ±0.03 g/cm³. Examples 4-6 are identical to Examples 1-3,except that the wait time of the pulse sequence was reduced to 2seconds. Examples 7-12 include ρ_(ma) and φ_(nmr) with largeruncertainties. The uncertainties σ(φ_(t)) in TABLE 2 range between 1.3and 1.9 p.u. and the uncertainties σ(V_(g),xo) range between 2.0 and 2.9p.u. Calculated φ_(density), φ_(t), V_(g),xo, and S_(g),xo, and erroranalysis σ(φ_(t)) and σ(V_(g),xo) are shown in TABLE 3.

Note that the uncertainties in φ_(t) and V_(g),xo are relatively smallconsidering there exist uncertainties in eight of the inputs.

                                      TABLE 2                                     __________________________________________________________________________    Synthetic Inputs for High-Porosity Shaly Gas Sand                             Ex.                                                                              ρ.sub.b                                                                        ρ.sub.ma                                                                       ρ.sub.f                                                                        ρ.sub.g                                                                        T.sub.1,gas                                                                        (HI).sub.g                                                                         (HI).sub.f                                                                         φ.sub.nmr                                                                       WT                                __________________________________________________________________________    1  2.2 ± 0.01                                                                      2.65 ± 0.0                                                                      1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 ± 1.0                                                                       0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.10 ± 0.01                                                                      4.0                               2  2.2 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 ± 1.0                                                                       0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.15 ± 0.01                                                                      4.0                               3  2.2 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 ± 1.0                                                                       0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.20 ± 0.01                                                                      4.0                               4  2.2 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 ± 1.0                                                                       0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.10 ± 0.01                                                                      2.0                               5  2.2 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 ± 1.0                                                                       0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.15 ± 0.01                                                                      2.0                               6  2.2 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 ± 1.0                                                                       0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.20 ± 0.01                                                                      2.0                               7  2.2 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 ± 1.0                                                                       0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.10 ± 0.01                                                                      4.0                               8  2.2 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 ± 1.0                                                                       0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.15 ± 0.01                                                                      4.0                               9  2.2 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 ± 1.0                                                                       0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.20 ± 0.01                                                                      4.0                               10 2.2 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 ± 1.0                                                                       0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.10 ± 0.01                                                                      2.0                               11 2.2 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 ± 1.0                                                                       0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.15 ± 0.01                                                                      2.0                               12 2.2 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 ± 1.0                                                                       0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.20 ± 0.01                                                                      2.0                               __________________________________________________________________________

                  TABLE 3                                                         ______________________________________                                        Synthetic Outputs for High-Porosity Shaly Gas Sand                            Ex.    φ.sub.density                                                                     φ.sub.t                                                                           V.sub.g                                                                            S.sub.g,xo                                                                          σ(φ.sub.t)                                                                σ(V.sub.g,xo)                   ______________________________________                                        1      0.27    0.205   0.14 0.69  0.013 0.020                                 2      0.27    0.224   0.10 0.44  0.013 0.021                                 3      0.27    0.244   0.06 0.24  0.014 0.023                                 4      0.27    0.210   0.13 0.62  0.013 0.017                                 5      0.27    0.228   0.09 0.41  0.013 0.019                                 6      0.27    0.246   0.06 0.22  0.015 0.022                                 7      0.27    0.205   0.14 0.68  0.019 0.027                                 8      0.27    0.224   0.10 0.44  0.018 0.027                                 9      0.27    0.244   0.06 0.24  0.019 0.029                                 10     0.27    0.210   0.13 0.62  0.018 0.024                                 11     0.27    0.228   0.09 0.41  0.018 0.025                                 12     0.27    0.246   0.06 0.22  0.019 0.027                                 ______________________________________                                    

LOW-POROSITY SHALY GAS SAND EXAMPLES

TABLES 4 and 5 are analogous to TABLES 2 and 3, except that the twelveexamples are for a low-porosity shaly gas sand formation. These examplesalso assume realistic uncertainties in the input variables.

Again, the uncertainties in φ_(t) and V_(g),xo are relatively smallconsidering there exist uncertainties in eight of the inputs. However,gas volume uncertainties σ(V_(g),xo) listed in TABLE 5 are of the sameorder of magnitude as gas volume uncertainties V_(g),xo, which meansthat gas volume quantification is more difficult in low-porosity zones.

                                      TABLE 4                                     __________________________________________________________________________    Synthetic Data Inputs for Low-Porosity Shaly Gas Sand                         Ex.                                                                              ρ.sub.b                                                                        ρ.sub.ma                                                                       ρ.sub.f                                                                        ρ.sub.g                                                                        T.sub.1,gas                                                                        (HI).sub.g                                                                         (HI).sub.f                                                                         φ.sub.mar                                                                       WT                                __________________________________________________________________________    1  2.5 ± 0.01                                                                      2.65 ± 0.0                                                                      1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 ± 1.0                                                                       0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.07 ± 0.01                                                                      4.0                               2  2.5 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 + 1.0                                                                          0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.06 ± 0.01                                                                      4.0                               3  2.5 ± 0.0l                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 + 1.0                                                                          0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.05 ± 0.01                                                                      4.0                               4  2.5 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 + 1.0                                                                          0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.07 ± 0.01                                                                      2.0                               5  2.5 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 + 1.0                                                                          0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.06 ± 0.01                                                                      2.0                               6  2.5 ± 0.0l                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 + 1.0                                                                          0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.05 ± 0.01                                                                      2.0                               7  2.5 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 + 1.0                                                                          0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.07 ± 0.015                                                                     4.0                               8  2.5 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 + 1.0                                                                          0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.06 ± 0.015                                                                     4.0                               9  2.5 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 + 1.0                                                                          0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.05 ± 0.015                                                                     4.0                               10 2.5 ± 0.0l                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 + 1.0                                                                          0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.07 ± 0.015                                                                     2.0                               11 2.5 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 + 1.0                                                                          0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.06 ± 0.015                                                                     2.0                               12 2.5 ± 0.01                                                                      2.65 + 0.0                                                                         1.0 ± 0.1                                                                       0.2 ± 0.1                                                                       4.0 + 1.0                                                                          0.4 ± 0.1                                                                       1.0 ± 0.1                                                                       0.05 ± 0.015                                                                     2.0                               __________________________________________________________________________

                  TABLE 5                                                         ______________________________________                                        Synthetic Outputs for Low-Porosity Shaly Gas Sand                             Ex.    φ.sub.density                                                                     φ.sub.t                                                                           V.sub.g,xo                                                                         S.sub.g                                                                             σ(φ.sub.t)                                                                σ(V.sub.g,xo)                   ______________________________________                                        1      0.09    0.083   0.017                                                                              0.21  0.012 0.018                                 2      0.09    0.079   0.025                                                                              0.32  0.012 0.018                                 3      0.09    0.075   0.033                                                                              0.44  0.012 0.017                                 4      0.09    0.083   0.016                                                                              0.19  0.012 0.017                                 5      0.09    0.080   0.023                                                                              0.29  0.012 0.016                                 6      0.09    0.076   0.031                                                                              0.41  0.012 0.016                                 7      0.09    0.083   0.017                                                                              0.21  0.019 0.027                                 8      0.09    0.079   0.025                                                                              0.32  0.019 0.027                                 9      0.09    0.075   0.033                                                                              0.44  0.019 0.027                                 10     0.09    0.083   0.016                                                                              0.19  0.019 0.025                                 11     0.09    0.080   0.023                                                                              0.29  0.019 0.025                                 12     0.09    0.076   0.031                                                                              0.41  0.019 0.025                                 ______________________________________                                    

It will be understood that the foregoing is only illustrative of theprinciples of the invention, and that various modifications can be madeby those skilled in the art without departing from the scope and spiritof the invention. For example, Equations (8), (15), and (16) assume thatwait time WT is sufficiently long to appreciably polarize the liquidphase. If this is not the case, these equations can be modified byreplacing every occurrence of (HI)_(f) by the product (HI)_(f) ·P_(f),where P_(f) is a polarization function. Furthermore, if the well isdrilled with oil-based mud and the reservoir is at an irreducible watersaturation, then φ_(nmr) can be corrected for insufficient wait time byapplying an oil-base mud filtrate polarization correction factor to thefree-fluid porosity. The corrected φ_(nmr) can be then be usedanalogously in Equations (8), (15), and (16). Although corrections forinsufficient polarization of the mud filtrate can be applied, a jobplanner is preferably used before logging to help select a wait timethat ensures sufficient polarization of the filtrate.

Furthermore, determining, calculating, solving, or using any of theequations or mathematical relationships included herein may be performedwith a commercially available processor downhole or uphole, as desired.

We claim:
 1. A method for characterizing a gas-bearing formationtraversed by a borehole comprising:computing an NMR-derived totalporosity φ_(nmr) and a density-derived total porosity φ_(density) ofsaid flushed zone; determining a gas-corrected total porosity φ_(t)using said φ_(nmr) and said φ_(density) ; determining a gas-correctedtotal water saturation S_(xot) of said flushed zone using said φ_(nmr)and said φ_(density) ; and determining a resistivity parameter using:##EQU39## where n is a saturation exponent, m is a model-dependentcementation exponent, C_(xo) is a conductivity of said flushed zone,C_(xo) being equal to 1/R_(xo), where R_(xo) is a flushed zoneresistivity, C_(mf) is a mud filtrate conductivity, C_(mf) being equalto 1/R_(mf), where R_(mf) is a resistivity of said mud filtrate, andwhere X is a model-dependent clay conductivity.
 2. The method of claim 1further comprising measuring a first portion of said earth formationwith a nuclear magnetic resonance tool to obtain NMR data.
 3. The methodof claim 2 further comprising measuring a second portion of said earthformation with a density tool to obtain density data.
 4. The method ofclaim 3 wherein said first portion and said second portion aresubstantially the same.
 5. The method of claim 1 wherein saiddetermining said φ_(t) comprises: ##EQU40## where (HI)_(g) is a HydrogenIndex of a gas, (HI)_(f) is a Hydrogen Index of a liquid phasecomprising mud filtrate and formation water, P_(g) is a gas polarizationfunction, which is defined as 1-exp (-WT/T_(l),gas), where WT is a waittime for a pulse sequence and T_(l),gas is a gas longitudinal relaxationtime at said condition, ##EQU41## where ρ_(f) is a density of saidliquid phase, ρ_(ma) is a formation matrix density, ρ_(g) is a densityof said gas, and ##EQU42## where ρ_(b) is a formation bulk density. 6.The method of claim 5 for use in complex lithology wherein said methodfurther comprises:providing lithology-dependent data to determine avolume V_(i) of a formation constituent i; and determining said matrixdensity ρ_(ma) using said lithology-dependent data substantiallyaccording to: ##EQU43## where ρ_(i) is a density of said constituent i.7. The method of claim 1 wherein said determining said φ_(t) issubstantially according to: ##EQU44## where said w is determinedsubstantially according to: ##EQU45## where (HI)_(g) is a Hydrogen Indexof a gas, (HI)_(f) is a Hydrogen Index of a liquid phase comprising mudfiltrate and formation water, P_(g) is a gas polarization function,which is defined as 1-exp (-WT/T_(l),gas), where WT is a wait time for apulse sequence and T_(l),gas is a gas longitudinal relaxation time,##EQU46## where ρ_(f) is a density of said liquid phase, ρ_(ma) is aformation matrix density, ρ_(g) is a density of said gas, and ##EQU47##where ρ_(b) is a formation bulk density.
 8. The method of claim 1 foruse in complex lithology wherein said method further comprises:providinglithology-dependent data to determine a volume V_(i) of a formationconstituent i; and determining said matrix density ρ_(ma) using saidlithology-dependent data substantially according to: ##EQU48## whereρ_(i) is a density of said formation constituent i.
 9. The method ofclaim 1 wherein said determining said S_(xot) is substantially accordingto: S_(xot) =V_(xot) /φ_(t), where V_(xot) is a total water volume ofsaid flushed zone.
 10. The method of claim 9 wherein said determiningsaid V_(xot) is substantially according to: V_(xot) =φ_(t) -V_(gxo),V_(gxo) is a gas volume of said flushed zone.
 11. The method of claim 10wherein said V_(g),xo is determined substantially according to:##EQU49##
 12. The method of claim 9 wherein said V_(g),xo is determinedsubstantially according to: V_(g),xo =S_(g),xo ·φ_(t), where S_(g),xo isa flushed zone gas saturation determined substantially according to: 13.The method of claim 1 wherein said determining a resistivity parametercomprises calculating a parameter selected from a group consisting ofsaid m, said X, and a cation exchange capacity per unit total porevolume Q_(v).
 14. The method of claim 13 wherein said determining aresistivity parameter comprises using a Waxman-Smits model.
 15. Themethod of claim 14 wherein said using a Waxman-Smits model comprisesusing X=Q_(v) B, where where C_(mf) is a conductivity of water, T_(C) isa temperature of said formation in degrees Celsius, and ##EQU50## whereF_(ws) is a Waxman-Smits formation factor,

    m.sub.ws =1.8167+1.6094 (1-e.sup.-1.2528y),

and ##EQU51##
 16. The method of claim 13 wherein said determining aresistivity parameter comprises using a Dual Water model.
 17. The methodof claim 16 wherein said using a Dual Water model comprises using:

    X=(C.sub.wb -C.sub.mf)S.sub.wb,

where C_(mf) is a conductivity of mud-filtrate, ##EQU52## where C_(wb)is a clay bound water conductivity, S_(wb) is a clay bound watersaturation, T_(C) is a temperature in degrees Celsius of said formation,and ##EQU53## where F_(dw) is a Dual Water formation factor,

    m.sub.dw =1.7762+0.3364(1-e.sup.-5.5035y)

and ##EQU54##
 18. The method of claim 13 wherein said determining aresistivity parameter further comprises: determining a true formationconductivity C_(true) in said virgin zone; anddetermining a virgin zonewater saturation S_(wt) substantially according to: ##EQU55## where saidC_(w) is a conductivity of water in said virgin zone.
 19. The method ofclaim 18 wherein said determining C_(true) in said virgin zone isperformed using a deep-resistivity measuring tool.
 20. The method ofclaim 19 further comprising determining C_(w) using an electrochemicalpotential model and SP logging data.
 21. The method of claim 20 furthercomprising determining a hydrocarbon saturation S_(hy) in said virginzone substantially according to: S_(hy) =1-S_(wt).
 22. A method forcharacterizing a formation traversed by a borehole comprising:receivingNMR data characterizing a flushed zone, said NMR data at leastcomprising P(T₂), which is a T₂ distribution; determining a clay boundwater volume V_(bound) substantially according to: ##EQU56## where T₂min is a minimum T₂ for clay bound water and T₂ max is a maximum T₂ forclay bound water; and determining a cation exchange capacity per unittotal pore volume Q_(v) using a clay bound water saturation S_(wb)model.
 23. The method of claim 22 further comprising determining saidS_(wb) substantially according to: S_(wb) =V_(bound) /φ_(t), where φ_(t)is a hydrocarbon corrected total porosity of said formation.
 24. Themethod of claim 23 wherein said φ_(t) is derived from φ_(nmr) andφ_(density).
 25. The method of claim 23 wherein said φ_(t) issubstantially determined according to: ##EQU57## where (HI)_(g) is aHydrogen Index of a gas, (HI)_(f) is a Hydrogen Index of a liquid phasecomprising mud filtrate and formation water, P_(g) is a gas polarizationfunction, which is defined as 1-exp (-WT/T_(l),gas), where WT is a waittime for a pulse sequence and T_(l),gas is a gas longitudinal relaxationtime at said condition, ##EQU58## where ρ_(f) is a density of saidliquid phase, ρ_(ma) is a formation matrix density, ρ_(g) is a densityof said gas, and ##EQU59## where ρ_(b) is a formation bulk density. 26.The method of claim 23 wherein said determining said Q_(v) issubstantially according to: Q_(v) =S_(wb) /α V_(q), where α is a Gouyexpansion factor of a clay diffuse layer and V_(q) is determinedsubstantially according to: V_(q) =4.97e-06 T_(c) ² -1.94e-03 T_(c)+0.342.
 27. The method of claim 26 wherein said α is determinedsubstantially according to: ##EQU60## where γ is an activity coefficientand said n is a salinity.
 28. The method of claim 27 wherein said n isdetermined substantially according to: ##EQU61## where said ppk is asalinity and ρ_(f) is a liquid phase density.
 29. The method of claim 27wherein said γ is determined substantially according to: ##EQU62## wheresaid coefficients a₁, a₂, a₃, a₄, and a₅, are determined substantiallyaccording to:

    a.sub.i =b.sub.i T.sub.c.sup.3 +c.sub.i T.sub.c.sup.2 +d.sub.i T.sub.c +e.sub.i,

where i=1, 2, 3, 4, and 5, respectively, m is a salinity of solventdetermined substantially according to: ##EQU63## and where saidcoefficients b_(i), c_(i), d_(i), and e_(i) are about:

    ______________________________________                                        i   b.sub.i    c.sub.i    d.sub.i  e.sub.i                                    ______________________________________                                        1   -6.1237e-11                                                                              +3.6490e-08                                                                              -1.2225e-06                                                                            +9.7432e-04,                               2   -3.1529e-08                                                                              +8.7540e-06                                                                              -1.3528e-03                                                                            -2.4460e-01,                               3   +1.5951e-08                                                                              -7.0447e-06                                                                              +1.0840e-03                                                                            +1.0514e-01,                               4   -1.0729e-09                                                                              +5.5435e-07                                                                              -1.0211e-04                                                                            +4.7400e-04, and                           5   +4.1937e-09                                                                              -2.1167e-06                                                                              +1.1317e-04                                                                            -3-6126e-02.                               ______________________________________                                    


30. The method of claim 22 wherein said determining said Q_(v) comprisesusing a Hill-Shirley-Klein model substantially according to: ##EQU64##where φ_(t) is a total porosity of said formation, and n is a salinity.31. A method for characterizing a formation traversed by a boreholecomprising:receiving SP logging data; and determining a cation exchangecapacity per unit total pore volume Q_(v) or a resistivity R_(w) usingan electrochemical potential model and said SP logging data.
 32. Themethod of claim 31 wherein said using said SP logging data comprisessolving ∇σ_(c) ·∇V=0, where σ_(c) is a conductivity and V is a potentialeverywhere in space, said solving comprising using at least a firstboundary condition and a second boundary condition, said first boundarycondition being V₂ -V₁ =SSP and said second boundary condition being J₂-J₁ =0, where SSP is a strength of an electro-chemical potential at aninterface between a flushed zone and a virgin zone, J is an electriccurrent density at said interface, and where subscripts 1 and 2 denotesaid flushed and virgin zones at said interface, respectively.
 33. Themethod of claim 32 wherein said solving comprises calculating said SSPusing a method selected from a group consisting of a finite elementmethod and a deconvolution method.
 34. The method of claim 33 whereinsaid =calculating said SSP further comprises measuring a spontaneouspotential SP formed between two points in said borehole, a resistivityof said flushed zone, a resistivity in said virgin zone, a position ofsaid electrochemical potential, a mud resistivity in said borehole, anda borehole cross-sectional area.
 35. The method of claim 34 wherein saidmethod of determining said SSP is substantially according to: ##EQU65##where k is a Boltzmann constant, T is an absolute temperature of saidformation, e is an electron charge, m is a salinity, γ is an activitycoefficient, C₊ is a cation conductivity and C₋ is an anionconductivity, and C is a rock conductivity determined substantiallyaccording to: C=C₊ +C₋.
 36. The method of claim 35 wherein said cationconductivity C₊ is determined substantially according to: ##EQU66## saidanion conductivity C₋ is determined substantially according to:

    C.sub.- =S.sub.xot.sup.n φ.sub.t.sup.m (1-t)C.sub.f,

where t is a cation transference number and C_(f) is a conductivity ofsaid fluid at said interface.
 37. The method of claim 36 wherein saidtransference number t is determined substantially according to:

    t=0.374-0.125 log (m)-1.77e-3 log.sup.2 (m)+4.047e-4(T.sub.C.sup.0 -25)-8.22e-7(T.sub.C.sup.0 -25).sup.2,

and C_(f) is determined substantially according to: ##EQU67##
 38. Themethod of claim 35 wherein said γ is determined substantially accordingto: where said coefficients a₁, a₂, a₃, a₄, and a₅, are determinedsubstantially according to:

    a.sub.i =b.sub.i T.sub.c.sup.3 +c.sub.i T.sub.c.sup.2 +d.sub.i T.sub.c +e.sub.i,

where i=1, 2, 3, 4, and 5, respectively, m is a salinity of solventdetermined substantially according to:

    ______________________________________                                         ##STR1##                                                                     where said coefficients b.sub.i, c.sub.i, d.sub.i, and e.sub.i are            about:                                                                        i    b.sub.i   c.sub.i    d.sub.i e.sub.i                                     ______________________________________                                        1    -6.1237e-11                                                                             +3.6490e-08                                                                              -1.2225e-06                                                                           +9.7432e-04,                                2    -3.1529e-08                                                                             +8.7540e-06                                                                              -1.3528e-03                                                                           -2.4460e-01,                                3    +1.5951e-08                                                                             -7.0447e-06                                                                              +1.0840e-03                                                                           +1.0514e-01,                                4    -1.0729e-09                                                                             +5.5435e-07                                                                              -1.0211e-04                                                                           +4.7400e-04, and                            5    +4.1937e-09                                                                             -2.1167e-06                                                                              +1.1317e-04                                                                           -3-6126e-02.                                ______________________________________                                    


39. A method for determining a bound fluid volume BFV of a formationwith complex lithology that is traversed by a boreholecomprising:receiving NMR data characterizing a flushed zone of saidformation, said NMR data at least comprising P(T₂), which is a T₂distribution; and determining a bound fluid volume BFV, said method ofdetermining said BFV comprising summing BFV_(i) constituents weighted bytheir respective constituent volumes V_(i), where i is an index denotingdifferent constituents.
 40. The method of claim 39 wherein saiddetermining said BFV is substantially according to: ##EQU68## where T₂min is a minimum T₂, and T₂ cutoff i is a cutoff T₂ of constituent i.41. The method of claim 40 further comprising determining a permeabilityk of said formation substantially according to a permeability model. 42.The method of claim 40 further comprising determining a T₂ cutoffsubstantially according to: ##EQU69## where T₂ min is a minimum T₂ andBFV is a known bound fluid volume for the formation.
 43. A method ofanalyzing the uncertainty of a gas-corrected petrophysical parameter,said method comprising calculating said variance of said parametersubstantially according to: ##EQU70## where ƒ is a petrophysicalparameter that is a function of n variables x_(n), σ² (ƒ) is a varianceof ƒ, and x_(i) * is a best estimate for each of said n variables. 44.The method of claim 43 wherein said ƒ is an output selected from thegroup consisting of gas-corrected total porosity φ_(t), a gas volume ofthe flushed zone V_(g),xo, and a flushed zone gas saturation S_(g),xo,and wherein said calculating uses quantities N₁, N₂, α⁰, and D, saidquantities being defined substantially according to: ##EQU71## where(HI)_(g) is a Hydrogen Index of a gas, (HI)_(f) is a Hydrogen Index of afluid, P_(g) is a gas polarization function, which is defined as 1-exp(-WT/T_(l),gas), where WT is a wait time for a pulse sequence andT_(l),gas is a gas longitudinal relaxation time at said condition,##EQU72## where ρ_(f) is a density of said fluid, ρ_(ma) is a formationmatrix density, ρ_(g) is a density of said gas, φ_(nmr) is a NMR-derivedporosity, and φ_(density) is a density-derived porosity determinedsubstantially according to: ##EQU73## where ρ_(b) is a formation bulkdensity.
 45. The method of claim 44 wherein said calculating saidvariance of said parameter comprises calculating said variance of saidφ_(t) using: ##EQU74##
 46. The method of claim 44 wherein saidcalculating said variance of said parameter comprises calculating saidvariance of said V_(g),xo using:
 47. The method of claim 44 wherein saidcalculating said variance of said parameter comprises calculating saidvariance of said S_(g),xo substantially according to: